The Szeged Index and Padmakar-Ivan Index on the Zero-Divisor Graph of a Commutative Ring

The zero-divisor graph of a commutative ring is a graph where the vertices represent the zero-divisors of the ring, and two distinct vertices are connected if their product equals zero. This study focuses on determining general formulas for the Szeged index and the Padmakar-Ivan index of the zero-divisor graph for specific commutative rings. The results show that for the first case of ring, the Szeged index is exactly half of the Padmakar-Ivan index. For the second case, the Szeged index is consistently greater than the Padmakar-Ivan index. These findings enhance the understanding of how the algebraic structure of rings influences the topological properties of their associated graphs.